A word problem becomes easier to solve when the child can turn the language into a representation that preserves the relationship. The purpose of drawing is not decoration. The purpose of a number bond is not to satisfy a template. A representation earns its place when it makes the mathematical structure easier to see, reason about and check.
This laboratory extends Word Problems, Mathematical Language, Representation and Reasoning, Addition and Subtraction Word-Problem Structures, Visual Models, Bar Models, Part–Whole Diagrams and Representation Choice, and the Number Line Laboratory.
The representation should expose the relationship before the operation is chosen.
A Five-Step Representation Routine
- Read the whole situation.
- Name what is known and unknown.
- Classify the relationship: combine, change, compare, missing part or equal groups.
- Choose a representation that makes that relationship visible.
- Write and solve the equation, then return to the story.
This routine is slower than keyword hunting at first, but it creates a reusable structure for unfamiliar questions.
Session 1 | Pictures for Concrete Combine Stories
Story: There are 6 red balloons and 4 blue balloons. How many balloons are there altogether?
Draw six marks for the red balloons and four marks for the blue balloons. The two groups are parts of one whole. A number bond can then record parts 6 and 4, whole 10. The equation is 6 + 4 = 10.
The drawing is useful because the child can point to the two parts before calculating.
Worked Example 1 | Combine
Eight books are on one shelf and five are on another. How many books are there altogether?
- Known parts: 8 and 5.
- Unknown whole: total books.
- Representation: number bond or part–whole bar.
- Equation: 8 + 5 = 13.
- Answer: 13 books.
Session 2 | Before-and-After Pictures for Change Stories
Story: Mei has 9 stickers. She gets 3 more. How many does she have now?
Use a before-and-after picture or number line. Start at 9, then show an increase of 3. The new amount is 12. The equation is 9 + 3 = 12.
For a decrease story, the movement goes the other way. The representation should show what changed and what the quantity became.
Worked Example 2 | Decrease
There are 14 birds. Five fly away. A number line can start at 14 and move backward 5 to 9. The equation is 14 − 5 = 9.
Session 3 | Aligned Bars for Comparison
Story: Kai has 12 marbles. Mei has 8 marbles. How many more does Kai have?
Draw two bars starting at the same left edge. Make Kai’s bar represent 12 and Mei’s represent 8. The unmatched part of Kai’s bar represents the difference. The equation is 12 − 8 = 4.
This representation prevents a common mistake: adding 12 and 8 merely because both numbers appear in the story.
Worked Example 3 | “Has More Than”
Mei has 14 beads. She has 4 more beads than Kai. How many beads does Kai have?
Draw Mei’s longer bar as 14. Mark a 4-unit extra segment beyond Kai’s shorter bar. The shorter amount is 14 − 4 = 10. Kai has 10 beads.
Session 4 | Number Bonds for Missing Parts
Story: A box contains 13 pencils. Eight are red and the rest are blue. How many are blue?
The whole is 13. One part is 8. A number bond shows the missing part directly. The equation can be 8 + □ = 13 or 13 − 8 = □. The missing part is 5.
This is not a take-away event even though subtraction can find the answer. The underlying structure is whole and parts.
Worked Example 4 | Missing Part
There are 16 children. Nine are wearing blue. The rest are wearing red. Whole: 16. Known part: 9. Missing part: 7. Therefore 16 − 9 = 7.
Session 5 | Unknown Start With a Timeline
Story: A basket had some apples. Four apples were added. Now there are 13. How many were there at first?
Draw a before box labelled ?, an arrow labelled +4, and an after box labelled 13. Working backwards gives 13 − 4 = 9. The starting quantity was 9.
The word “added” describes the event, but subtraction finds the unknown start. This is why operation-by-keyword rules are fragile.
Worked Example 5 | Unknown Start After a Decrease
A tray had some counters. Five were removed. Fourteen remain. The before amount must contain both the 14 remaining and the 5 removed. Therefore 14 + 5 = 19.
Session 6 | Choose Between Representations
Give three stories with the same numbers 12 and 7:
- 12 counters, 7 removed — change/decrease.
- 12 red and 7 blue — combine.
- 12 stickers versus 7 stickers — compare.
Ask which representation makes each relationship easiest to see. The same surface numbers do not determine the same operation or model.
Representation Choice Matrix
| Problem structure | Useful representations |
|---|---|
| combine | picture, number bond, part–whole bar |
| increase/decrease | before–after drawing, number line, timeline |
| compare | aligned bars, paired objects, number-line distance |
| missing part | number bond, part–whole bar |
| unknown start | before–after diagram, timeline, inverse equation |
| equal groups | objects, repeated-addition picture, array |
A Representation Can Be Correct but Unhelpful
A child could draw 27 individual dots for a simple 20 + 7 place-value problem. The drawing may be mathematically valid but inefficient. A tens-and-ones model would expose the structure more clearly.
Teach learners to ask not only “Is my drawing correct?” but “Does this representation make the relationship easier to see?”
Pictures Must Match the Story
If a problem says 9 birds and 4 more arrive, a picture showing four birds crossed out contradicts the event. The arithmetic might later be repaired, but the first weak link is already in the representation.
Bar Models Need Labels
Two unlabeled rectangles can be ambiguous. Label the quantities or people they represent and mark the unknown. At Primary 1, labels can be short: Mei 12, Kai 8, difference ?.
Labels turn a generic drawing into a representation of this particular problem.
Number Lines Need a Starting Point
For 9 + 4, mark 9 before making four forward jumps. A child who begins counting at 9 as jump one may land on 12 instead of 13.
The line should distinguish position from change.
Equations Come After the Relationship
Once the representation is clear, record the equation. In a comparison story with 12 and 8, the equation 12 − 8 = 4 describes the visible unmatched part. The sign is chosen because of the relationship, not because the word “more” appeared.
Worked Example 6 | One Story, Two Valid Representations
Story: 7 children are on a bus. 5 more get on. A before–after picture can show 7 becoming 12. A number line can start at 7 and jump forward 5. Both preserve the increase relationship.
Ask the learner which is easier to check and why. Representation choice is part of mathematical control.
Extra Information and Missing Information
Represent only information relevant to the mathematical relationship. If a story says a green basket contains 8 apples and 5 oranges and asks for the fruit total, the basket colour need not appear in the mathematical model.
If the whole is unknown and only one part is supplied, no representation can invent the missing data. The correct response may be “not enough information”.
Common Representation Errors
| Error | Likely weak link |
|---|---|
| draws every detail from story | relevance filtering weak |
| uses part–whole model for a change without showing time order | problem structure not differentiated |
| adds comparison quantities | aligned difference not understood |
| number line starts from zero unnecessarily | starting quantity not identified |
| bar model unlabeled | representation detached from context |
| operation chosen before representation | keyword or habit driving strategy |
Twenty Practice Questions
- There are 6 red balls and 4 blue balls. Which representation would you choose and what is the total?
- Nine birds are joined by 3 more. Show the change and find the new total.
- Fourteen birds are present and 5 fly away. Find the result with a number line or before–after model.
- Mei has 12 stickers and Kai has 8. How many more does Mei have?
- A box has 13 pencils. Eight are red. How many are not red?
- A basket had some apples. Four were added and now there are 13. Find the start.
- A tray had some counters. Five were removed and 14 remain. Find the start.
- There are 8 boys and 7 girls. How many children altogether?
- One ribbon is 15 cm and another is 11 cm. Find the difference.
- A class has 16 pupils. Nine choose apples. The rest choose bananas. How many choose bananas?
- Which representation best shows a comparison: number bond, aligned bars, or random picture?
- Which representation best shows four equal groups of 3?
- Which representation best shows 27 + 8 using a bridge through 30?
- Which representation best shows a known whole and one missing part?
- True or false: every word problem containing “more” should use addition.
- True or false: a correct equation can come from an incorrect picture.
- Explain why labels improve a bar model.
- A story gives 5 visible counters but no total and asks how many are hidden. Is there enough information?
- Create a combine story for 8 + 5 = 13.
- Create a compare story for 13 − 5 = 8.
Explained Answers
1. 10. A picture, number bond or part–whole bar can show the two parts combining. 2. 12. Start with 9 and add 3. 3. 9. Move backward 5 from 14. 4. 4. Aligned bars or number-line distance make the comparison visible.
5. 5. Whole 13, known part 8. 6. 9. Work backward from 13 by 4. 7. 19. Rebuild the start from 14 remaining plus 5 removed. 8. 15. Combine 8 and 7. 9. 4 cm. Compare the two lengths.
10. 7. Missing part of 16. 11. Aligned bars. They show matched and unmatched quantities. 12. Objects or an array. Equal groups should be visible. 13. An open number line. Jump 3 to 30, then 5. 14. A number bond or part–whole bar.
15. False. “Has 4 more than” may require subtraction when the smaller quantity is unknown. 16. Yes. Arithmetic can accidentally be correct even when the model does not match the story; the representation still needs repair. 17. Labels identify what each bar and unknown represent. 18. No. The hidden amount is not determined without a whole or another sufficient condition. 19–20. Many stories are valid if they preserve the requested relationship.
A Strong Practice Progression
- Act out concrete stories.
- Draw simple pictures that preserve actions or groups.
- Use number bonds for whole-and-parts.
- Use number lines for change and distance.
- Use aligned bars for comparison.
- Move the unknown to different positions.
- Choose among two possible representations.
- Reject irrelevant details.
- Recognise when information is insufficient.
- Write equations only after the relationship is visible.
What Adults Can Ask
- “What is happening in the story?”
- “What do we know?”
- “What must we find?”
- “Which picture or model makes that relationship visible?”
- “What does each part of your model stand for?”
- “Does your equation match your representation?”
Return to the Primary 1 Mathematics Learning Hub for the complete route.
